Sunday, September 24, 2006

Linearity Property of the Integral

The linearity property of the integral refers to the very important property of being able to break integrals apart into sums. That is, ∫(b,a) [f(x)dx + g(x)dx] = ∫ (b,a) f(x)dx + ∫(b,a)g(x)dx.

The content in today's blog is taken from Edwards and Penney's Calculus and Analytic Geometry.

Theorem: Linearity Property of Integrals

If α, β are constants and f(x) and g(x) are continuous functions on [a,b], then:

∫ (b,a) [αf(x) + βg(x)]dx = α∫(b,a)f(x)dx + β∫(b,a)g(x)dx


(1) By the Constant Multiple Property (see Lemma 2, here):

∫(b,a) cf(x)dx = c∫(b,a) f(x)dx

(2) Let F(x) be the antiderivative of f(x) and G(x) be the antiderivative of g(x).

(3) d/dx[F(x) + G(x)] = f(x) + g(x) [See Lemma 3, here]

(4) Using the Fundamental Theorem of Calculus (see Thereom 2, here), this gives us:

∫ (b,a) [f(x) + g(x)]dx = [F(x) + G(x)] (b,a)

(5) Using the Evaluation of Integrals (see Theorem 3, here), we have:

[F(x) + G(x)](b,a) = [F(b) + G(b)] - [F(a) + G(a)] = [F(b) - F(a)] + [G(b) - G(a)] =

= [F(x)](b,a) + [G(x)](b,a) = ∫(b,a) f(x)dx + ∫(b,a) g(x)dx.



1 comment :

Timothy Chen Allen said...

Your post on the linearity property of the integral really just helped me. I'm studying Mathematical Statistics and this term was used in a proof that the Expected Value function is linear. It was good to see this worked out. Thanks!